Constant Sequence in Topological Space Converges

Theorem

Let $X$ be a topological space.

Let $x \in X$.

Define a sequence $\sequence {x_n}_{n \in \N}$ by:

$x_n = x$ for each $n \in \N$.

Then $\sequence {x_n}_{n \in \N}$ converges to $x$.

Proof

Let $U$ be an open neighborhood of $x$.

Then we have $x_n = x \in U$ for all $n \in \N$.

Hence $\sequence {x_n}_{n \in \N}$ converges to $x$.

$\blacksquare$